Genome-wide analyses of sparse mediation effects under composite null hypotheses

A genome-wide mediation analysis is conducted to investigate whether epigenetic variations M mediate the effect of socioeconomic adversity S on adiposity Y . The mediation effect can be expressed as a product of two parameters: the S-M association and the M -Y association conditional on S. We show that the joint significance test examining the two parameters separately has smaller p-values than the normality-based or the normal product-based test for the product and is a size α test. However, under multiple tests with sparse signals, the conventional joint significance test has a conservative test size and low power within a study because of the sparsity in signals and not accounting for the composition of different null hypotheses. We develop a novel test assessing the product of two normally distributed test statistics under a composite null hypothesis, where either one parameter is zero or both are zero. We show that the null composition can be adjusted by variances of test statistics without directly estimating proportions of different nulls. Advantages of the new test are illustrated in simulation and the epigenomic study. The new test identifies four methylation loci mediating the socioeconomic effect on adiposity with the false discovery rate less than 20% while existing methods had none surviving this cut-off.